On {a, b}-Edge-Weightings of Bipartite Graphs with Odd a, b

For any S ⊂ ℤ we say that a graph G has the S-property if there exists an S-edge-weighting w : E(G) → S such that for any pair of adjacent vertices u, v we have ∑_e∈E(v) w(e) ≠ ∑_e∈E(u) w(e), where E(v) and E(u) are the sets of edges incident to v and u, respectively. This work focuses on {a, a+2}-edge-weightings where a ∈ ℤ is odd. We show that a 2-connected bipartite graph has the {a, a+2}-property if and only if it is not a so-called odd multi-cactus. In the case of trees, we show that only one case is pathological. That is, we show that all trees have the {a, a+2}-property for odd a ≠ −1, while there is an easy characterization of trees without the {−1, 1}-property.